Segmentation of trajectories on nonmonotone criteria

Boris Aronov, Anne Driemel, Marc Van Kreveld, Maarten Löffler, Frank Staals

    Research output: Contribution to journalArticle

    Abstract

    In the trajectory segmentation problem, we are given a polygonal trajectory with n vertices that we have to subdivide into a minimum number of disjoint segments (subtrajectories) that all satisfy a given criterion. The problem is known to be solvable efficiently for monotone criteria: criteria with the property that if they hold on a certain segment, they also hold on every subsegment of that segment. To the best of our knowledge, no theoretical results are known for nonmonotone criteria. We present a broader study of the segmentation problem, and suggest a general framework for solving it, based on the start-stop diagram: a 2-dimensional diagram that represents all valid and invalid segments of a given trajectory. This yields two subproblems: (1) computing the start-stop diagram, and (2) finding the optimal segmentation for a given diagram. We show that (2) is NP-hard in general. However, we identify properties of the start-stop diagram that make the problem tractable and give a polynomial-time algorithm for this case. We study two concrete nonmonotone criteria that arise in practical applications in more detail. Both are based on a given univariate attribute function f over the domain of the trajectory. We say a segment satisfies an outlier-tolerant criterion if the value of f lies within a certain range for at least a given percentage of the length of the segment. We say a segment satisfies a standard deviation criterion if the standard deviation of f over the length of the segment lies below a given threshold. We show that both criteria satisfy the properties that make the segmentation problem tractable. In particular, we compute an optimal segmentation of a trajectory based on the outlier-tolerant criterion in O(n2 log n + kn2) time and on the standard deviation criterion in O(kn2) time, where n is the number of vertices of the input trajectory and k is the number of segments in an optimal solution.

    Original languageEnglish (US)
    Article number26
    JournalACM Transactions on Algorithms
    Volume12
    Issue number2
    DOIs
    StatePublished - Dec 1 2015

    Fingerprint

    Segmentation
    Trajectory
    Diagram
    Standard deviation
    Outlier
    Subdivide
    Polynomial-time Algorithm
    Univariate
    Percentage
    Monotone
    Disjoint
    NP-complete problem
    Optimal Solution
    Attribute
    Valid
    Computing
    Range of data

    Keywords

    • Dynamic programming
    • Geometric algorithms
    • Segmentation
    • Trajectory

    ASJC Scopus subject areas

    • Mathematics (miscellaneous)

    Cite this

    Aronov, B., Driemel, A., Van Kreveld, M., Löffler, M., & Staals, F. (2015). Segmentation of trajectories on nonmonotone criteria. ACM Transactions on Algorithms, 12(2), [26]. https://doi.org/10.1145/2660772

    Segmentation of trajectories on nonmonotone criteria. / Aronov, Boris; Driemel, Anne; Van Kreveld, Marc; Löffler, Maarten; Staals, Frank.

    In: ACM Transactions on Algorithms, Vol. 12, No. 2, 26, 01.12.2015.

    Research output: Contribution to journalArticle

    Aronov, B, Driemel, A, Van Kreveld, M, Löffler, M & Staals, F 2015, 'Segmentation of trajectories on nonmonotone criteria', ACM Transactions on Algorithms, vol. 12, no. 2, 26. https://doi.org/10.1145/2660772
    Aronov B, Driemel A, Van Kreveld M, Löffler M, Staals F. Segmentation of trajectories on nonmonotone criteria. ACM Transactions on Algorithms. 2015 Dec 1;12(2). 26. https://doi.org/10.1145/2660772
    Aronov, Boris ; Driemel, Anne ; Van Kreveld, Marc ; Löffler, Maarten ; Staals, Frank. / Segmentation of trajectories on nonmonotone criteria. In: ACM Transactions on Algorithms. 2015 ; Vol. 12, No. 2.
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